Let $X$ be a compact metric space, and let $K_1,K_2,\dots$ be a sequence of subsets of $X$ which are nonempty, closed, and connected, and which satisfy $K_{n+1}\subset K_n$ for every positive integer $n$. Prove that intersection of $K_n$ for $n$ from $1$ to infinity is connected.
So, this was c...
@DanielFischer I wonder. This is probably easy, I just haven't thought about it. One can define a polynomial in $R[x_1,\ldots,x_r]$ to be homogeneous if all its constituent monomials have the same total degree. But we can also define them as those for which $P(t{\bf x})=t^p P({\bf x})$ for some constant $p$ (the total degree of the monomials). The former implies the latter easily, can we show the latter implies the former without much problem?
@PedroTamaroff There are two kinds of "false-positives" that can occur using the equation $P(t\mathbf x)=t^pP(\mathbf x)$. One is based on the fact that there could conceivably exist a number $n$ such that $a^n=a$ for all $a\in R$. The other is that you can lose information going from a formal polynomial in $R[\mathbf x]$ to a polynomial function. In particular my guess that it would work for all integral domains $R$ is wrong.
Problem: Evaluate $\zeta(4)$ using the Poisson summation formula. I'm thinking of finding the Fourier transform of $\frac{1}{(x^2 + a^2)^2}$ by convolving the transforms of $\frac{1}{x^2 + a^2}$ and then taking the limit as $a \to 0$. The sum over the convolved Fourier transforms should be some a sum over some kind of exponential which I'm hoping can be massaged into something trigonometric
after meta.stackoverflow started throwing errors, now math.SE says "This site is currently in read-only mode; we’ll return with full functionality soon." Looks like this is the end.
@EnjoysMath Just out of curiosity, are rings related to something you and usukidoll were discussing before, or did you just think "hey, I'm going to tell someone what a ring is now"? I mean this in the best possible way.
@MarkS. Did you know that $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A \approx A(p)$, where $A$ is a finite abelian group and $A(p)$ is it's Sylow $p$-subgroup?
Proof that 1 is not in the union? Basically yes. But it's not proof that the union contains all of (-1,1).
@Sush Some advice for things like math.stackexchange.com/questions/578990 . If you're not satisfied with an answer, you can upvote it but then not accept it, so that other people know the question hasn't been answered to your satisfaction.
(or not upvote it if you don't think it's worth it)
@Sush However, you should also try to follow the guidelines found here: meta.math.stackexchange.com/a/1804/26369 if this is a homework question/textbook question. Explain what your thoughts have been in detail, etc.
@MarkS. Sorry. Now I have made it unaccepted. I will surely keep your advice in my mind for later questions.(I am not native English speaker, so please pardon me if I am wrong in communication skill. ) This was not any homework, but I was self reading Mathmatics for Economists by Simon and Blume.
Don't worry about your English. What you've said has been clear enough. If you edit your question to say "I was self reading Mathmatics for Economists by Simon and Blume." it will help people write answers close to your level/style.
So, I'm trying to plan my lecture for determinants for a college algebra class, and it occurs to me that I still don't really understand what a determinant tells us, or some interesting applications of it. I could go with finding area of parallelograms/pipeds, but is there anything else that would possibly interest a college algebra class?
Prove the statement for the case $n = 1$. This will be the base case for an induction.
Let $R = \Bbb{Z}[e_1, \dots, e_{n-1}], \ S = R[e_n]$. When refering to $R$ we're refering to the isomorphic copy of $R$ in $S$. So we have $R \lt S$ as rings.
Let $M = \Bbb{Z}[x_1, \dots, x_{n-1}]$, and $N...
Please check. Kthx
And get back to me, I really want to know if that trick worked
Could you please give some upvote to the following?http://math.stackexchange.com/questions/556977/gaussian-integrals-over-a-half-space Just to give it some attention! Thanks!
Hi. If anyone is good with regular language, please tell me for each one if those languages are regular or not (I think they're all irregular, and proved with pumping lemma): $\{ a^ky \mid y \in \Sigma^* , |y|_a \leq k \}$ $\{ a^ky \mid y \in \Sigma^* , |y|_a \geq k \}$ $\{ 0^n1^k2^m \mid m=max(n,k) \}$
@Zeezee Sometimes it takes a while before someone who can answer your question sees it. If a few days go by and you still don't have an answer, you might think of adding a bounty.
@PedroTamaroff Basically, the integral is elementary, but maybe you think of a fast way. You may firstly factorize $x^2$ in denominator. Then, by symmetry you only consider the integral over half of the interval.
Suppose that $f(z)=z+a_2 z^2+a_3 z^3+ \dots$. Why then we can say that $$\frac{1}{f(z)}=\frac{1}{z}+c_0+c_1 z + c_2 z^2 + \dots$$ for some $c_i \in \mathbb{C}, i \in \mathbb{N}$ ?
And I don't understand way nobody can help me with this my question: math.stackexchange.com/questions/578158/…. I was thinking that this is elementary, but.... I will give bounty if nobody answer.
@EnjoysMath You really have to let the arrows lead the way. There is not much you can do. Pick an element of $b'\in B'$. Where can you go? To $\phi'(b')=c\in C'$ by $\phi'$. But you know $\gamma$ is surjective, so you get back to $C$. And so on...
@PedroTamaroff I have taken most introductory courses, basic number theory, basic functional analysis, some abstract algebra and some real analysis. Also a fair bit of linear algebra.
My choices are many.. Galois theory, Manifolds, Complex Analysis, General Topology, Analytic Numbertheory, Classical Mechanics etc.
Let $X$ be a compact metric space, and let $K_1,K_2,\dots$ be a sequence of subsets of $X$ which are nonempty, closed, and connected, and which satisfy $K_{n+1}\subset K_n$ for every positive integer $n$. Prove that intersection of $K_n$ for $n$ from $1$ to infinity is connected.
So, this was c...
Prove the statement for the case $n = 1$. This will be the base case for an induction.
Let $R = \Bbb{Z}[e_1, \dots, e_{n-1}], \ S = R[e_n]$. When refering to $R$ we're refering to the isomorphic copy of $R$ in $S$. So we have $R \lt S$ as rings.
Let $M = \Bbb{Z}[x_1, \dots, x_{n-1}]$, and $N...
